Equicontinuous families

Let \((X,d_X)\) and \((Y,d_Y)\) be metric spaces, let \(x_0\in X\), and let \(\mathcal{F}\) be a family of functions \(X\to Y\).

• The family \(\mathcal{F}\) is equicontinuous at \(x_0\) if for every \(\varepsilon > 0\) there exists \(\delta=\delta(\varepsilon, x_0)>0\) such that for all \(x\in X\) and all \(f\in\mathcal{F}\), \[ x\in B_\delta(x_0) \implies f(x) \in B_\varepsilon(f(x_0)). \]

• The family \(\mathcal{F}\) is uniformly equicontinuous if for every \(\varepsilon>0\) there exists \(\delta=\delta(\varepsilon)>0\) such that for all \(x_1,x_2\in X\) and all \(f\in\mathcal{F}\), \[ x_1\in B_\delta(x_2) \implies f(x_1) \in B_\varepsilon(f(x_2)). \]